Kubo-Martin-Schwinger,Non-Equilibrium Thermal states,

and Conformal Field Theory

Roberto Longo

Paul Martin Memorial

Harvard, October 2016

Based on a joint work with S. Hollands

and previous works with Bischoff, Kawahigashi, Rehren and Camassa, Tanimoto, Weiner

Thermal equilibrium states

Thermodynamics concerns heat and temperature and their relationto energy and work. A primary role is played by the equilibriumdistribution.

Gibbs states

Finite quantum system: A matrix algebra with Hamiltonian H andevolution τt = Ade itH . Equilibrium state ϕ at inverse temperatureβ is given by the Gibbs property

ϕ(X ) =Tr(e−βHX )

Tr(e−βH)

What are the equilibrium states at infinite volume where there isno trace, no inner Hamiltonian?

The Kubo-Martin-Schwinger condition

The fundamental KMS equilibrium condition originated by thestudy of the analytic properties of the Green functions in twopapers in the 50’s, one by Kubo, one by Paul Martin and J.Schwinger.

R. Kubo (1957), “Statistical-Mechanical Theory of IrreversibleProcesses. I. General Theory and Simple Applications to Magneticand Conduction Problems”, Journal of the Physical Society ofJapan 12, 570-586

Paul C. Martin, Julian Schwinger (1959), “Theory ofMany-Particle Systems. I”, Physical Review 115, 1342-1373,

The final form was presented by Haag, Hugenholtz and Winnink atthe 1967 Baton Rouge conference on Operator Algebras.

KMS states

Infinite volume. A a C ∗-algebra, τ a one-par. automorphism groupof A. A state ϕ of A is KMS at inverse temperature β > 0 if forX ,Y ∈ A ∃ FXY ∈ A(Sβ) s.t.

(a) FXY (t) = ϕ(X τt(Y )

)(b) FXY (t + iβ) = ϕ

(τt(Y )X

)where A(Sβ) is the algebra of functions analytic in the stripSβ = 0 < =z < β, bounded and continuous on the closure Sβ.

(Note: it is sufficient to check (a) and (b) for X ,Y in a dense∗-subalgebra B.)

KMS states have been so far the central objects in the QuantumStatistical Mechanics, for example in the analysis of phasetransition.

Modular theory

Let M be a von Neumann algebra and ϕ a normal faithful state onM. The fundamental Tomita-Takesaki theorem states that thereexists a canonical one-parameter automorphism group σϕ of Massociated with ϕ. So, von Neumann algebras are intrinsicallydynamical objects.

In the GNS representation ϕ = (Ω, ·Ω); if S is the closure of

xΩ 7→ x∗Ω, with polar decomposition S = J∆12 , we have

σϕt = Ad∆it :∆itM∆−it =M, JMJ = M ′

For a remarkable historical coincidence, Tomita announced thetheorem at the 1967 Baton Rouge conference. Soon later Takesakicharcterised the modular group by the KMS condition.

Some highlights concerning KMS/modular theory

Modular/KMS theory is fundamental in Operator Algebras. Inparticular, it allowed the classification of type III factors by AlainConnes, leading him to the Fields Medal.

Later crucial use (among others):

I KMS and stabilty (Haag-Kastler-Trych Polhmeyer)

I Passivity (Putz-Woronowicz)

I QFT, Bisognano-Wichmann theorem and Hawking-Unruheffect (Sewell)

I Positive cones - noncommutative measure theory (Araki,Connes, Haagerup)

I Non-commutative Chern character in cyclic cohomology(Jaffe-Lesniewski-Osterwalder)

I Jones index in the type III setting (Kosaki, L.)

Non-equilibrium thermodynamics

Non-equilibrium thermodynamics: study physical systems not inthermodynamic equilibrium but basically described by thermalequilibrium variables. Systems, in a sense, near equilibrium; but, ingeneral, non-uniform in space and time.

Non-equilibrium thermodynamics has been effectively studied fordecades with important achievements, yet the general theory is stillmissing. The framework is even more incomplete in the quantumcase, non-equilibrium quantum statistical mechanics.

We aim provide a general, model independent scheme for the abovesituation in the context of quantum, two dimensional ConformalQuantum Field Theory. As we shall see, we provide the generalpicture for the evolution towards a non-equilibrium steady state.

A typical frame described by Non-Equilibrium Thermodynamics:

R1

β1

probe. . . . . . R2

β2

Two infinite reservoirs R1, R2 in equilibrium at their owntemperatures T1 = β−1

1 , T2 = β−12 , and possibly chemical

potentials µ1, µ2, are set in contact, possibly inserting a probe.

As time evolves, the system should reach a non-equilibrium steadystate.

This is the situation we want to analyse. As we shall see theOperator Algebraic approach to CFT provides a model independentdescription, in particular of the asymptotic steady state, and exactcomputation of the expectation values of the main physicalquantities.

Non-equilibrium steady states

A non-equilibrium steady state NESS ϕ of A satisfies property (a)in the KMS condition, for all X ,Y in a dense ∗-subalgebra of B,but not necessarily property (b).

For any X ,Y in B the function

FXY (t) = ϕ(X τt(Y )

)is the boundary value of a function holomorphic in Sβ. (Ruelle)

Example: the tensor product of two KMS states at temperaturesβ1, β2 is a NESS with β = min(β1, β2).

Problem: describe the NESS state ω and show that the initialstate ψ evolves towards ω

limt→+∞

ψ · τt = ω

Mobius covariant nets (Haag-Kastler nets on S1)A local Mobius covariant net A on S1 is a map

I ∈ I → A(I ) ⊂ B(H)

I ≡ family of proper intervals of S1, that satisfies:

I A. Isotony. I1 ⊂ I2 =⇒ A(I1) ⊂ A(I2)

I B. Locality. I1 ∩ I2 = ∅ =⇒ [A(I1),A(I2)] = 0I C. Mobius covariance. ∃ unitary rep. U of the Mobius group

Mob on H such that

U(g)A(I )U(g)∗ = A(gI ), g ∈ Mob, I ∈ I.

I D. Positivity of the energy. Generator L0 of rotation subgroupof U (conformal Hamiltonian) is positive.

I E. Existence of the vacuum. ∃! U-invariant vector Ω ∈ H(vacuum vector), and Ω is cyclic for

∨I∈I A(I ).

Consequences

I Irreducibility:∨

I∈I A(I ) = B(H).

I Reeh-Schlieder theorem: Ω is cyclic and separating for eachA(I ).

I Bisognano-Wichmann property (KMS property of ω|A(I )):The modular operator/conjugation ∆I and JI of (A(I ),Ω) are

U(δI (2πt)) = ∆−itI , t ∈ R, dilations

U(rI ) = JI reflection

(Frohlich-Gabbiani, Guido-L.)

I Haag duality: A(I )′ = A(I ′)

I Factoriality: A(I ) is III1-factor (in Connes classification)

I Additivity: I ⊂ ∪i Ii =⇒ A(I ) ⊂ ∨iA(Ii ) (Fredenhagen,Jorss).

Local conformal netsDiff(S1) ≡ group of orientation-preserving smooth diffeomorphisms of S1.

Diff I (S1) ≡ g ∈ Diff(S1) : g(t) = t ∀t ∈ I ′.

A local conformal net A is a Mobius covariant net s.t.

F. Conformal covariance. ∃ a projective unitary representation Uof Diff(S1) on H extending the unitary representation of Mob s.t.

U(g)A(I )U(g)∗ = A(gI ), g ∈ Diff(S1),

U(g)xU(g)∗ = x , x ∈ A(I ), g ∈ Diff I ′(S1),

−→ unitary representation of the Virasoro algebra

[Lm, Ln] = (m − n)Lm+n +c

12(m3 −m)δm,n

−→ stress-energy tensor:

T (z) =∑n∈Z

Lnz−n−2

Representations

A (DHR) representation ρ of local conformal net A on a Hilbertspace H is a map I ∈ I 7→ ρI , with ρI a normal rep. of A(I ) on Hs.t.

ρI A(I ) = ρI , I ⊂ I , I , I ⊂ I .

Index-statistics relation (L.):

d(ρ) =[ρI ′(A(I ′)

)′: ρI(A(I )

)] 12

DHR dimension =√

Jones index

Complete rationality (Kawahigashi, Muger, L.)

µA ≡[(A(I1) ∨ A(I3)

)′:(A(I2) ∨ A(I4)

)]<∞

=⇒µA =

∑i

d(ρi )2

A is modular. (Later work by Xu, L. and Morinelli, Tanimoto,Weiner removed extra requirements.)

Circle and real line picture

-1

∞

P P'

z 7→ iz − 1

z + 1

We shall frequently switch between the two pictures.

KMS and Jones index

Kac-Wakimoto formula (conjecture)

Let A be a conformal net, ρ representations of A, then

limt→0+

Tr(e−tL0,ρ)

Tr(e−tL0)= d(ρ)

Analog of the Kac-Wakimoto formula (theorem)

ρ a representation of A:

(ξ, e−2πKρξ) = d(ρ)

where Kρ is the generator of the dilations δI and ξ is any vectorcyclic for ρ(A(I ′)) such that (ξ, ρ(·)ξ) is the vacuum state onA(I ′).

Basic conformal nets

U(1)-current net

Let A be the local conformal net on S1 associated with theU(1)-current algebra. In the real line picture A is given by

A(I ) ≡ W (f ) : f ∈ C∞R (R), supp f ⊂ I′′

where W is the representation of the Weyl commutation relations

W (f )W (g) = e−i∫fg ′W (f + g)

associated with the vacuum state ω

ω(W (f )) ≡ e−||f ||2, ||f ||2 ≡

∫ ∞0

p|f (p)|2dp

where f is the Fourier transform of f .

W (f ) = exp(− i

∫f (x)j(x)dx

),[j(f ), j(g)

]= i

∫fg ′dx

with j(x) the U(1)-current.

There is a one parameter family γq, q ∈ R of irreducible sectorsand all have index 1 (Buchholz, Mack, Todorov)

γq(W (f )) ≡ e i∫Ff W (f ), F ∈ C∞,

1

2π

∫F = q .

q is the called the charge of the sector.

Virasoro nets

For every possible value of the central charge c, let Uc theirreducible rep. of Diff(S1) with lowest weight zero:

Virc(I ) ≡ Uc(DiffI (S1))′′

Virc is contained in every conformal net.

A classification of KMS states (Camassa, Tanimoto,Weiner, L.)

How many KMS states do there exist?

Completely rational case

A completely rational: only one KMS state (geometricallyconstructed) β = 2πexp: net on R A → restriction of A to R+

exp A(I ) = AdU(η)

η diffeomorphism, ηI = exponential

geometric KMS state on A(R) = vacuum state on A(R+) exp

ϕgeo = ω exp

Note: Scaling with dilation, we get the geometric KMS state atany give β > 0.

Non-rational case: U(1)-current model

The primary (locally normal) KMS states of the U(1)-current netare in one-to-one correspondence with real numbers q ∈ R;

Geometric KMS state: ϕgeo = ϕ0

Any primary KMS state:

ϕq = ϕgeo γq.

whereγq(W (f )) = e iq

∫f (x)dxW (f )

γq is equivalent to the BMT q-sector.

Virasoro net: c = 1

(With c < 1 there is only one KMS state: the net is completelyrational)

Primary KMS states of the Vir1 net are in one-to-onecorrespondence with positive real numbers |q| ∈ R+; each stateϕ|q| is uniquely determined by its value on the stress-energytensor T :

ϕ|q| (T (f )) =

(π

12β2+

q2

2

)∫f dx .

The geometric KMS state corresponds to q = 0, and the

corresponding value of the ‘energy density’ π12β2 + q2

2 is the lowestone in the set of the KMS states.

(We construct these KMS states by composing the geometric statewith automorphisms on the larger U(1)-current net.)

Virasoro net: c > 1

There is a set of primary (locally normal) KMS states of the Vircnet with c > 1 w.r.t. translations in one-to-one correspondencewith positive real numbers |q| ∈ R+; each state ϕ|q| can beevaluated on the stress-energy tensor

ϕ|q| (T (f )) =

(π

12β2+

q2

2

)∫f dx

and the geometric KMS state corresponds to q = 1β

√π(c−1)

6 andenergy density πc

12β2 .

Are they all? Probably yes...

Chemical potential

A a local conformal net on R (or on M) and ϕ an extremalβ-KMS state on A w.r.t. the time translation group τ and ρ anirreducible DHR localized endomorphism of A ≡ ∪I⊂RA(I ) withfinite dimension d(ρ). Assume that ρ is normal, namely it extendsthe weak closure M of A; automatic e.g. if ϕ satisfies essentialduality πϕ

(A(I±)

)′ ∩M = πϕ((A(I∓)

)′′, I± the ±half-line.

U time translation unitary covariance cocycle in A:

AdU(t) · τt · ρ = ρ · τt , t ∈ R ,

with U(t + s) = U(t)τt(U(s)

)(cocycle relation) (unique by a

phase, canonical choice by Mob covariance).

U is equal up to a phase to a Connes Radon-Nikodym cocycle:

U(t) = e−i2πµρ(ϕ)td(ρ)−iβ−1t(Dϕ · Φρ : Dϕ

)−β−1t

.

µρ(ϕ) ∈ R is the chemical potential of ϕ w.r.t. the charge ρ.

Here Φρ is the left inverse of ρ, Φρ · ρ = id, so ϕ · Φρ is a KMSstate in the sector ρ.

The geometric β-KMS state ϕ0 has zero chemical potential.

By the holomorphic property of the Connes Radon-Nikodymcocycle:

e2πβµρ(ϕ) = anal. cont.t−→ iβ

ϕ(U(t)

)/anal. cont.

t−→ iβϕ0

(U(t)

).

Example, BMT sectors:

With ϕβ,q the β-state associated withe charge q, the chemicalpotential w.r.t. the charge q is given by

µp(ϕβ,q) = qp/π

By linearity µp is determined at p = 1, so we may putµ(ϕβ,q) = q/π.

2-dimensional CFT

M = R2 Minkowski plane.(T00 T10

T01 T11

)conserved and traceless stress-energy tensor.

As is well known, T+ = 12 (T00 + T01) and T− = 1

2 (T00 − T01) arechiral fields,

T+ = T+(t + x), T− = T−(t − x).

Left and right movers.

Ψk family of conformal fields on M: Tij + relatively local fieldsO = I × J double cone, I , J intervals of the chiral lines t ± x = 0

A(O) = e iΨk (f ), suppf ⊂ O′′

then by relative locality

A(O) ⊃ AL(I )⊗AR(J)

AL,AR chiral fields on t ± x = 0 generated by TL,TR and otherchiral fields

(completely) rational case: AL(I )⊗AR(J) ⊂ A(O) has finiteJones index.

Phase boundaries (Bischoff, Kawahigashi, Rehren, L.)

ML ≡ (t, x) : x < 0, MR ≡ (t, x) : x > 0 left and right halfMinkowski plane, with a CFT on each half.

Transmissive solution for left/right chiral stress energy tensors:

T L+(t) = TR

+ (t), T L−(t) = TR

− (t).

,A transpartent phase boundary is given by specifying two localconformal nets BL and BR on ML/R on the same Hilbert space H;

ML ⊃ O 7→ BL(O) ; MR ⊃ O 7→ BR(O) ,

BL and BR both contain a common chiral subnet A = A+ ⊗A−.so BL/R extends on the entire M by covariance. By causality:[

BL(O2),BR(O1)]

= 0, O1 ⊂ ML, O2 ⊂ MR , O1 ⊂ O ′2

By diffeomorphism covariance, BR is thus right local w.r.t. BL

Given a phase boundary, we consider the von Neumann algebras

D(O) ≡ BL(O) ∨ BR(O) , O ∈ K .

D is in general non-local, but relatively local w.r.t. A. D(O) mayhave non-trivial center. In the completely rational case,A(O) ⊂ D(O) has finite Jones index, so the center of D(O) isfinite dimensional; we may cut down by a minimal projection of thecenter (a defect) and assume D(O) to be a factor.

Universal construction (and classification) is done byconsidering the braided product of the Q-systems associated withA+ ⊗A− ⊂ BL and A+ ⊗A− ⊂ BR .

Cf. Frohlich, Fuchs, Runkel, Schweigert (Euclidean setting)

Non-equilubrium states in CFT (S. Hollands, R.L.)Two local conformal nets BL and BR on the Minkowski plane M,both containing the same chiral net A = A+ ⊗A−. For themoment BL/R is completely rational, so the KMS state is unique,later we deal wih chemical potentials.

Before contact. The two systems BL and BR are, separately, each

in a thermal equilibrium state. KMS states ϕL/RβL/R

on BL/R at

inverse temperature βL/R w.r.t. τ , possibly with βL 6= βR .

BL and BR live independently in their own half plane ML and MR

and their own Hilbert space. The composite system on ML ∪MR isgiven by

ML ⊃ O 7→ BL(O) , MR ⊃ O 7→ BR(O)

with C ∗-algebra BL(ML)⊗BR(MR) and state

ϕ = ϕLβL|BL(ML) ⊗ ϕR

βR|BR(MR) ;

ϕ is a stationary but not KMS.

After contact.

At time t = 0 we put the two systems BL on ML and BR on MR incontact through a totally transmissible phase boundary. We are inthe above phase boundary case, BL and BR are now nets on Macting on a common Hilbert space H; the algebras BL(WL) andBR(WR) commute.

We want to describe the state ψ of the global system after contactat time t = 0. As above, we set

D(O) ≡ BL(O) ∨ BR(O)

ψ should be a natural state on the global algebra D that satisfies

ψ|BL(WL) = ϕLβL|BL(WL), ψ|BR(WR) = ϕR

βR|BR(WR) .

Since BL(ML) and BR(MR) are not independent, the existence ofsuch state ψ is not obvious. Clearly the C ∗-algebra on Hgenerated by BL(WL) and BR(WR) is naturally isomorphic toBL(WL)⊗BR(WR) and the restriction of ψ to it is the productstate ϕL

βL|BL(WL) ⊗ ϕR

βR|BR(WR).

∃ a natural state ψ ≡ ψβL,βR on D s.t. ψ|B(WL/R) is ϕL/RβL/βR

(natural on D(x ± t 6= 0), our initial state.

The state ψ is given by ψ ≡ ϕ · αλL,λR , where ϕ is the geometricstate on D (at inverse temperature 1) and α = αλL,λR is the abovedoubly scaling automorphism with λL = β−1

L , λR = β−1R .

By inserting a probe ψ the state will be normal.

The large time limit. After a large time we expect the globalsystem to reach a non equilibrium steady state ω.

Let ϕ+βL

, ϕ−βR be the geometric KMS states respectively on A+ andA− with inverse temperature βL and βR ; we define

ω ≡ ϕ+βL⊗ ϕ−βR · ε ,

so ω is the state on D obtained by extending ϕ+βL⊗ϕ−βR from A to

D by the conditional natural expectation ε : D→ A. Clearly ω is astationary state, indeed:

ω is a NESS on D with β = minβL, βR.

Initial state ψ ≡ ψβL,βR Final state ω

Properties βL on W aL ,

βR on W bR ,

unspecified on W bL ∩W a

R

Determined on V+

Definition on chi-ral subnet

doubly scaling the geo-metric KMS state by di-lations

ϕ+βL⊗ ϕ−βR

Global definition again by doubly scaling extension by con-ditional expectationω = ϕ+

βL⊗ ϕ−βR · ε

We now want to show that the evolution ψ · τt of the initial stateψ of the composite system approaches the non-equilibrium steadystate ω as t → +∞.

Note that:ψ|D(O) = ω|D(O) if O ∈ K(V+)

We have:

For every Z ∈ D we have:

limt→+∞

ψ(τt(Z )

)= ω(Z ) .

Indeed, if Z ∈ D(O) with O ∈ K(M) and t > tO , we haveτt(Z ) ∈ D(V+) as said, so

ψ(τt(Z )

)= ω

(τt(Z )

)= ω(Z ) , t > tO ,

because of the stationarity property of ω.See the picture.

Case with chemical potential

We suppose here that A± in the net C contains is generated by theU(1)-current J± (thus BL/R is non rational with central chargec = 1).

Given q ∈ R, the β-KMS state ϕβ,q on D with charge q is definedby

ϕβ,q = ϕ+β,q ⊗ ϕ

−β,q · ε ,

where ϕ±β,q is the KMS state on A± with charge q.

ϕβ,q satisfies the β-KMS condition on D w.r.t. to τ .

Similarly as above we have:

Given βL/R > 0, qL/R ∈ R, there exists a state ψ on D such that

ψ|BL(WL) = ϕβL,qL |BL(WL) , ψ|BR(WR) = ϕβR ,qR |BR(WR) .

and for every Z ∈ D we have:

limt→+∞

ψ(τt(Z )

)= ω(Z ) .

We can explicitly compute the expected value of the asymptoticNESS state ω on the stress energy tensor and on the current(chemical potential enters):

Now ω = ϕ+βL,qL

⊗ ϕ−βR ,qR · ε is a steady state is a NESS and ω isdetermined uniquely by βL/R and the charges qL/R

ϕ+βL,qL

(J+(0)

)= qL , ϕ−βR ,qR

(J−(0)

)= qR .

We also have

ϕ+βL,qL

(T +(0)

)=

π

12β2L

+q2L

2, ϕ−βR ,qR

(T−(0)

)=

π

12β2R

+q2R

2.

In presence of chemical potentials µL/R = 1πqL/R , the large time

limit of the two dimensional current density expectation value(x-component of the current operator Jµ) in the state ψ is, withJx(t, x) = J−(t + x)− J+(t − x)

limt→+∞

ψ(Jx(t, x)

)= ϕ−βL,qL

(J−(0)

)−ϕ+

βR ,qR

(J+(0)

)= −π(µL−µR) ,

whereas on the stress energy tensor

limt→+∞

ψ(Ttx(t, x)

)= ϕ+

βL,qL

(T +(0)

)− ϕ−βR ,qR

(T−(0)

)=

π

12

(β−2L − β

−2R

)+π2

2

(µ2L − µ2

R

),

(cf. Bernard-Doyon)The above discussion could be extended to the case A± contains ahigher rank current algebra net.